Eigenvalue relations for decorated trivalent polyhedra - Connections between the fullerenes and their fulleren-yne and spheriphane relatives

The adjacency spectrum of a trivalent parent cage determines completely the spectra of a range of formally related frameworks. Analytical generating relations are given for vertex truncation, vertex insertion and ten combined decorations, leading to the prediction of pi stability for several series of molecular species. Expanded cages in which bonds are replaced by even-numbered carbon chains (e.g. fulleren-ynes), are predicted to have closed pi shells for a closed-shell parent, e.g. a leapfrog or truncated fullerene. Spheriphanes and related polycoronands, in which parent vertices are replaced by benzenoid rings, are predicted to have closed pi shells irrespective of the pi configuration of the parent.